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Check if a number N starts with 1 in b-base

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Given a number N and base b if N in base b representation starts with 1 print Yes else print No 

Examples :  

Input : n = 6, b = 4
Output : Yes
6 can be written as 12_4 in base 4so answer is Yes as it starts with 1Input : n = 24, b = 2Output : Yes24 can be written as 11000_2 in base 2so answer is Yes as it starts with 1Input :  n = 24, b = 7Output : No24 can be written as 33_7 in base 7so answer is No as it starts with 3        

When a number N is represented in base ‘b’ it gets converted to m+1 length sequence b_{m}      [Tex]b_{m-1}      [/Tex]…..b_0      which implies b_{m}      *b^{m}      b_{m-1}      *b^{m-1}      …..+b_0      *b^0      = N 

The smallest number in base b and starting with ‘1’ i.e. 100..00 and m+1 digits in base is b^{m}
and largest number is 2*b^{m}      -1.So N should lie in this range. 
b^{m}      <= N <= 2*b^{m}      -1 
Now another thing to notice is that m cannot exceed floor(log_2      (N)) because when we represent any number in base-2 it gets converted into a sequence of only 1s and 0s so the length of this sequence will always be greater than any other base representation and its length will be equal to floor(log_2      (N))+1. 

So to check for a given base ‘b’ if N starts with 1 or not we will traverse from m = 1 to m = floor(log_2      (N)) and check if for any m N lies in the range b^{m}      <= N <= 2*b^{m}      -1 or not and accordingly print “Yes” or “No”. 
 

C++

// C++ program to check if number starts with
// one in base b representation
#include <bits/stdc++.h>
using namespace std;
  
// Returns true if n starts with  1 in
// base b representation
bool CheckIfstartsWithOne(int n, int b)
{
    // highest m can be log2(n)
    int m = log2(n);
    for (int i = 1; i <= m; i++) {
        // if b^m <= N <= 2*b^m - 1,
        // return true
        if (n >= pow(b, i) && n <= 2 * pow(b, i) - 1)
            return true;
    }
    return false;
}
  
// printing yes or no
void printYesORno(int n, int b)
{
    if (CheckIfstartsWithOne(n, b) == true)
        cout << "Yes" << endl;
    else if (CheckIfstartsWithOne(n, b) == false)
        cout << "No" << endl;
}
  
// driver function
int main()
{
    printYesORno(6, 4);
    printYesORno(24, 2);
    printYesORno(24, 7);
    printYesORno(24, 15);
}

                    

Java

// Java program to check if number starts with
// one in base b representation
  
class GFG {
      
    // Returns true if n starts with 1 in
    // base b representation
    static boolean CheckIfstartsWithOne(int n, int b)
    {
          
        // highest m can be log2(n)
        int m = (int)(Math.log10(n) / Math.log10(2));
        for (int i = 1; i <= m; i++) {
              
            // if b^m <= N <= 2*b^m - 1,
            // return true
            if (n >= (int)Math.pow(b, i) &&
                n <= 2 * (int)Math.pow(b, i) - 1)
                return true;
        }
          
        return false;
    }
  
    // Driver method
    public static void main(String args[])
    {
        System.out.println(
           CheckIfstartsWithOne(6, 4) ? "Yes" : "No");
             
        System.out.println(
           CheckIfstartsWithOne(24, 2) ? "Yes" : "No");
             
        System.out.println(
           CheckIfstartsWithOne(24, 7) ? "Yes" : "No");
             
        System.out.println(
           CheckIfstartsWithOne(24, 15) ? "Yes" : "No");
    }
}

                    

Python3

# Python3 program to check 
# if number starts with one
# in base b representation
import math
  
# Returns true if n 
# starts with 1 in 
# base b representation
def CheckIfstartsWithOne(n, b):
      
    # highest m can be log2(n)
    m = (int)(math.log2(n));
    for i in range (1, m + 1):
          
        # if b^m <= N <= 2*b^m - 1,
        #return true
        x = (int)(math.pow(b, i));
        if n >= x and n <= 2 * x - 1:
            return 1;
    return 0;
  
# printing yes or no
def printYesORno(n, b):
    if CheckIfstartsWithOne(n, b) == 1:
        print("Yes");
    if CheckIfstartsWithOne(n, b) == 0:
        print("No");
          
# Driver Code
printYesORno(6, 4);
printYesORno(24, 2);
printYesORno(24, 7);
printYesORno(24, 15);
  
# This code is contributed by mits.

                    

C#

// C# program to check if number starts with
// one in base b representation
using System;
  
class GFG{
      
// Returns true if n starts with 1 in
// base b representation
static bool CheckIfstartsWithOne(int n, int b)
{
      
    // highest m can be log2(n)
    int m = (int)(Math.Log10(n) / Math.Log10(2));
    for(int i = 1; i <= m; i++) 
    {
          
        // if b^m <= N <= 2*b^m - 1,
        // return true
        if (n >= (int)Math.Pow(b, i) &&
            n <= 2 * (int)Math.Pow(b, i) - 1)
            return true;
    }
      
    return false;
}
  
// Driver code
public static void Main(String []args)
{
    Console.WriteLine(
       CheckIfstartsWithOne(6, 4) ? "Yes" : "No");
         
    Console.WriteLine(
       CheckIfstartsWithOne(24, 2) ? "Yes" : "No");
         
    Console.WriteLine(
       CheckIfstartsWithOne(24, 7) ? "Yes" : "No");
         
    Console.WriteLine(
       CheckIfstartsWithOne(24, 15) ? "Yes" : "No");
}
}
  
// This code is contributed by Princi Singh

                    

PHP

<?php
// PHP program to check if 
// number starts with one 
// in base b representation
  
// Returns true if n starts 
// with 1 in base b representation
function CheckIfstartsWithOne($n, $b)
{
    // highest m can be log2(n)
    $m = log($n, 2);
    for ($i = 1; $i <= $m; $i++) 
    {
        // if b^m <= N <= 2*b^m - 1,
        // return true
        if ($n >= pow($b, $i) && 
                $n <= 2 * pow($b, $i) - 1)
            return true;
    }
    return false;
}
  
// printing yes or no
function printYesORno($n, $b)
{
    if (CheckIfstartsWithOne($n, $b) == true)
        echo "Yes" ,"\n";
    else if (CheckIfstartsWithOne($n, $b) == false)
        echo "No" ,"\n";
}
  
// Driver Code
printYesORno(6, 4);
printYesORno(24, 2);
printYesORno(24, 7);
printYesORno(24, 15);
  
// This code is contributed by ajit 
?>

                    

Javascript

<script>
    // Javascript program to check if number starts
    // with one in base b representation
      
    // Returns true if n starts with
    // 1 in base b representation
    function CheckIfstartsWithOne(n, b)
    {
            
        // highest m can be log2(n)
        let m = parseInt(Math.log10(n) / Math.log10(2), 10);
            
        for (let i = 1; i <= m; i++)
        {
                
            // if b^m <= N <= 2*b^m - 1,
            // return true
            if (n >= Math.pow(b, i) && n <= 2 * Math.pow(b, i) - 1)
                return true;
        }
            
        return false;
    }
      
    document.write(CheckIfstartsWithOne(6, 4) ? "Yes" + "</br>" : "No" + "</br>");
               
    document.write(CheckIfstartsWithOne(24, 2) ? "Yes" + "</br>" : "No" + "</br>");
  
    document.write(CheckIfstartsWithOne(24, 7) ? "Yes" + "</br>" : "No" + "</br>");
  
    document.write(CheckIfstartsWithOne(24, 15) ? "Yes" : "No");
      
</script>

                    

Output : 

Yes
Yes
No
Yes

Time Complexity: O(m), where m is calculated as log2(n)
Auxiliary Space: O(1), as no extra space is required


 



Last Updated : 20 Feb, 2023
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